I simply can't see a situation where getting a JD makes sense. PE firms don't care about prestige for the sake of prestige - many firms these days don't require an MBA if you have a good trajectory. If you have good PE experience prior to school and you build a strong network you can position yourself to stay in the industry. ~15% go into PE from H/S, ~10% from W, ~5% from Booth/Columbia/Kellogg. Schools are pretty careful in their admissions criteria though and you will notice that the % coming into the industry pre-MBA and going back post-MBA is almost identical.
Unless you want to be a lawyer don't get a JD. If you want to drop half a million in tuition and opportunity cost get a JD/MBA but that is also quite marginal. Trust your skills and focus on building good deal exp. now, and that will yield good results in the admissions process.
Just my 2c